8. CONSTRAINTS ON COSMOLOGIES

The most important outcome of the newer experimental results is that the
standard cosmological paradigm is in good shape. A large amount of high
precision data on the power spectrum is adequately fit with fewer than 10
free parameters. The framework is that of Friedmann-Robertson-Walker models,
which have nearly flat geometry, containing Dark Matter and Dark Energy,
and with adiabatic perturbations having close to scale invariant initial
conditions.

Within this framework, bounds can be placed on the values of the
cosmological parameters. Of course,
much more stringent constraints can be placed on models
which cover a restricted number of parameters, e.g. assuming that
tot = 1,
n = 1 or r = 0. More generally, the
constraints depend upon the adopted priors, even if they are implicit,
for example by restricting the parameter freedom or the ranges of
parameters (particularly where likelihoods peak near the boundaries), or
by using different choices of other data in combination with the CMB.
When the data become even more precise,
these considerations will become less important,
but for now we caution that restrictions on model space and choice of
priors need to be kept in mind when adopting specific parameter values and
uncertainties.

There are some combinations of parameters that fit the CMB anisotropies
almost equivalently. For example, there is a nearly exact geometric
degeneracy, where any combination of
M and
that gives the same angular diameter distance to last
scattering will give nearly identical
Cs. There are also other
near degeneracies among the parameters. Such degeneracies can be broken
when using the CMB data in combination with other cosmological data sets.
Particularly useful are complementary constraints from galaxy clustering,
the abundance of galaxy clusters, weak gravitational lensing measurements,
Type Ia supernova distances and the distribution of Lyman
forest
clouds. For an overview of some of these other cosmological constraints,
see Ref. [12].

The combination of WMAP, CBI and ACBAR, together with weak priors
(on h and
Bh2 for example),
and within the context of a 6 parameter family of
models (which fixes
tot = 1),
yields the following results
[43]:
A = 2.7( ± 0.3) × 10-9,
n = 0.97 ± 0.03,
h = 0.73 ± 0.05,
Bh2 = 0.023 ± 0.001,
Mh2 = 0.13 ± 0.01 and
= 0.17 ± 0.07.
Note that for h, the CMB data alone provide only a very weak
constraint, unless spatial flatness or some other cosmological data are
used. For
Bh2 the precise value depends
sensitively on how much freedom is allowed in the shape of the primordial
power spectrum (see `Big-Bang Nucleosynthesis' mini-review
[44]).
For the optical depth ,
the error bar is large enough that apparently quite different results can
come from other combinations of data.

The best constraint on
tot is
1.02 ± 0.02. This comes from
including priors from h and supernova data. Slightly different, but
consistent results come from using different data combinations.

The 95% confidence upper limit on r is 0.53 (including some extra
constraint from galaxy clustering). This limit is stronger if we restrict
ourselves to n < 1 and weaker if we allow
dn / d ln k 0.

There are also constraints on parameters over and above the basic 8 that we
have described. But for such constraints it is necessary to include
additional data in order to break the degeneracies. For example the
addition of the Dark Energy equation of state, w adds the partial
degeneracy of being able to fit a ridge in (w, h) space,
extending to
low values of both parameters. This degeneracy is broken when the CMB is
used in combination with independent H0 limits, for
example
[45],
giving w < - 0.5 at 95% confidence. Tighter limits can be
placed using restrictive model-spaces and/or additional data.

For the optical depth ,
the error bar is large enough that apparently quite different results can
come from other combinations of data.
The constraint from the combined WMAPCTT and
CTE data is
= 0.17 ± 0.04,
which corresponds (within reasonable models) to a reionization redshift
9 < zi < 30 (95% CL)
[40].
This is a little higher than some theoretical predictions and some
suggestions from studies of absorption in high-z quasar spectra
[46].
The excitement here is that we have direct information from CMB polarization
which can be combined with other astrophysical measurements to understand
when the first stars formed and brought about the end of the cosmic dark
ages.